A364114 a(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^3 for n >= 0.
1, 7, 163, 5623, 235251, 11009257, 554159719, 29359663991, 1615702377331, 91558286583757, 5310712888211413, 313940484249068761, 18853030977961798359, 1147317139889540758509, 70618205829113737707663, 4389482803713232076789623, 275190242843266217113413491
Offset: 0
Examples
Examples of supercongruences: a(7) - a(1) = 29359663991 - 7 = (2^4)*(7^3)*37*144589 == 0 (mod 7^3). a(7 - 1) - a(0) = 554159719 - 1 = 2*(3^4)*(7^3)*9973 == 0 (mod 7^3). a(5^2) - a(5) = 5343160378366596176372561346633696195759257 - 11009257 = (2^4)*(5^6)*21372641513466384705490245386534784739 == 0 (mod 5^6). a(5^2 - 1) - a(5 - 1) = 81394273032250674032560324508765757297751 - 235251 = (2^2)*(5^6)*7*13*29*6317*78120239161449483411026081851 == 0 (mod 5^6).
Programs
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Maple
a(n) := coeff(series( 1/(1-x)* LegendreP(n,(1+x)/(1-x))^3, x, 21), x, n): seq(a(n), n = 0..20);
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Mathematica
Table[SeriesCoefficient[1/(1 - x) * LegendreP[n, (1 + x)/(1 - x)]^3, {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 09 2023 *)
Formula
a(n) ~ (1921 + 533*sqrt(13))^(n + 1/2) / (13^(1/4) * Pi^2 * n^2 * 2^(n + 7/2) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Jul 09 2023
Conjectures:
1) 3*a(p) - 11*a(p-1) == 10 (mod p^5) for all primes p >= 7 (checked up to p = 101).
2) a(p)^21 == (7^21)*a(p-1)^11 (mod p^5) for all primes p >= 7 (checked up to p = 101).
Comments